The Limits to Altruism - University of Oxfordusers.ox.ac.uk/~sedm1375/GPIpresentation.pdf · The Limits to Altruism Richard Povey Hertford College and St Hilda’s College, University

The Limits to Altruism

Richard Povey

Hertford College and St Hilda’s College, University of Oxford

May 18th 2018

Presentation given at the Global Priorities Institute, University of Oxford. Slides are

available at: users.ox.ac.uk/~sedm1375/GPIpresentation.pdf

Richard Povey The Limits to Altruism

users.ox.ac.uk/~sedm1375/GPIpresentation.pdf

Positive and Normative Limits to Altruism

Positive Limits - Evidence for Imperfect Altruism

Experimental / Behavioural Game TheoryEconomics of HappinessPolitical EconomyBiased Altruism Within the Family

Normative Limits - Social Optimality of Imperfect Altruism

Fundamental Theorems of Welfare EconomicsRotten Kid Theorem and Samaritan’s DilemmaPunishment in Dynamic GamesEvolutionary Theory

Normative Limits - Policy Consequences of ImperfectAltruism

Market failures and corrective policiesCost-benefit analysis


Rings of Altruism

The following diagram provides a visualisation of an individual’saltruistic linkages to other individuals. The inner right mightrepresent a family or proximate group.


Strong and Weak Reciprocity

Weak reciprocity is the form of altruism that can be seen asenlightened self-interest – individuals do each other a goodturn because they rationally expect to be “paid back”.However, this is insufficient to explain the complex functionalintegration of human societies [Fehr & Gachter, 2000].Strongly reciprocal altruism takes a positive and negativeform, where individuals either help or harm others at materialcost to themselves. This acts as “glue” holding institutionstogether, because the willingness of strong reciprocators topunish “cheats” forces selfish individuals to also behave well.The importance of the willingness of individuals to engage inaltruistic punishment has also been reflected in culturalselection theory. Altruistic punishment is a key mechanismwhich acts as an “altruism amplification device”, because it isusually less costly to punish another individual (e.g. byostracising them) than it is to make an altruistic sacrifice fortheir benefit [Sober & Wilson, 1999].


Experimental / Behavioural Game Theory

Some of the most important games that have been extensivelytested in laboratory environments are:

Finitely-repeated prisoners’ dilemmaPublic goods gamesDictator or ultimatum gamesCentipede game

There is a vast literature which it would be foolish to attemptto summarise here, but a strong consensus that thepredictions of classical game theory (based upon self interestand perfect common knowledge of rationality) aresystematically violated. There is also a consensus that in orderto explain observed behaviour it is necessary to introduce bothlimitations upon perfect common knowledge of rationality andother-regarding preferences.


Prisoners’ Dilemma

H L

Hb − c

b − c

b

−c

L−c

b

0

0

If agents play a high altruism strategy (H) then convey a benefit b upon theother player but themselves incur a cost c.

Since L is a strictly dominant strategy, backwards induction can be used to showthat any finitely-repeated prisoners’ dilemma results in a uniquesubgame-perfect Nash equilibrium with L played by both players in every period.

However, significant co-operation occurs in finitely-repeated experimentalprisoners’ dilemma games. Andreoni and Miller conclude on the basis ofexperimental evidence that rational reputation-building on the part of mostagents plus true altruistic behaviour on the part of a minority offers the bestexplanation for this phenomenon [Andreoni & Miller, 1993].


Public Goods Games

Public Goods games are similar to N-player prisoners’ dilemma but where eachplayer can choose how much to contribute, with each unit of contributioncreating a benefit b which is shared over the group but at a cost b > c > b

N.

Evidence [Dawes & Thaler, 1988] shows that for small groups averagecontributions are usually in the region of 40%-60% of the optimal level. Whenthe game is repeated with the same individuals playing, the average level ofcontributions tends to drop over time. However, the ability to altruisticallypunish non-co-operators and non-punishers greatly increases the ability tosustain co-operation [Fehr & Gachter, 2000] [Fehr & Fischbacher, 2003].


The Ultimatum Game

The “ultimatum game” is played between two individuals.The first individual proposes the division of £x between thetwo individuals and the second individual can either acceptthe offer or refuse, in which case both get a payoff of 0.

If both individuals are selfish and there is full commonknowledge of rationality, classical game theory predicts thatthe first individual offers the smallest amount they can that ishigher than 0 (i.e. 1p) and that the second individual accepts.

However, when the game is played in experimental situations,the predicted outcome occurs extremely rarely, and there issignificant variation between cultures regarding the amountthat the first individual offers to the second. The empiricalevidence has been summarised as showing that offers areusually between 30% and 40%, with the mode often being50%. Very few offers are below 20%, and those which are thislow are often rejected [Camerer & Thaler, 1995].


The Ultimatum Game

Andreoni et al. have extended the ultimatum game toconvexify the strategy space of the second individual byallowing them to continuously shrink the “pie” after theallocation is chosen by the proposer. Around 40% of subjectswere found to have convex preferences for equity as illustratedby the diagrams below, whilst around 50% were found to haveselfish preferences [Andreoni et al., 2003]:


Centipede Game

McKelvey and Palfrey conduct experimental centipede games and find thattypically players pass for a number of periods before somebody takes the largerpile [McKelvey & Palfrey, 1992]. They explain this using the idea that aproportion of the population is altruistic, and that selfish individuals can pretendto be altruistic in order to get their opponent to co-operate. By calibrating themodel to their data, they estimate that 5% of the population is believed to bealtruistic [McKelvey & Palfrey, 1992].


Economics of Happiness

A key result of this literature is that the positive relationshipbetween happiness and income is greater within a society than it isover time as a society develops. This strongly suggests thepresence of negative relative income effects (“keeping up with theJones’ ”) which would seem to have a connection with negativestrong reciprocity / preferences for fairness. [Easterlin, 1974][Clark et al., 2008] [Layard, 2006].


Biased Altruism within the Family

Research into the discount rates parents are revealed to apply to their children’swelfare in making costly decisions related to children’s health, as measured bylead contamination, has shown them to be similar to market interest rates forwealthier parents and higher for poorer parents, but nowhere near as high as thediscount rates of 20% - 50% found to be applied to consumer durables[Agee & Crocker, 1996]. This suggests a strong degree of parental altruism.

Researchers into differentials in wages and human capital investment betweenmales and females in the US [Behrman et al., 1986] have attempted todetermine whether or not this is driven by greater weight upon the success ofmale children in parents’ altruistic utility functions. The conclusion of this studywas that existing wage differentials reinforce inequalities in human capitalinvestment, but that parents do not, at root favour boys (in fact, if anything,the raw weighting on the welfare of girls is slightly higher). A similar study ofthe Phillipines [Davies & Zhang, 1995], however, found evidence for pure genderbias, underlining the fact that altruistic imperfections are culturally relative.

Evidence on the treatment of step children in the US [Case et al., 1999], on theother hand, suggests that they do receive a smaller proportion of family incomeon food if they live with a stepmother, after controlling for income.


Political Economy

Research on the relationship between environmental valuations reported insurvey data and life expectancy [Popp, 2001] has also found evidence of a rolefor partial altruism. If people are fully altruistic, their life expectancy should notaffect their contingent valuation of environmental goods. On the other hand, ifthey are fully selfish, the valuation should be, on average, zero as life expectancygoes to zero. The evidence, however, rejects both these hypotheses, suggestingthe presence of partial altruism. The central estimate is of an equal weightingbetween individual welfare and the average welfare of future generations, butthis estimate is very sensitive to the assumed discount rate.

Some of the most striking evidence for imperfections to altruism comes from therealm of international political economy. A 1998 study which sought to estimatethe marginal cost of additional life expectancy in different countries found thatthe implicit valuation of a life year in the richest countries was 300 times that inthe poorest countries. Once difference in average life expectancy are taken intoaccount, the cost of saving an entire lifespan in the richest countries came to1000 times that of the poorest countries [Dowrick et al., 1998].


Normative Limits

There is much confusion of the ideal that a person oughtto be allowed to pursue his own aims with the belief that,if left free, he will or ought to pursue solely his selfishaims.

[Hayek, 1960]

In my view the ideal society would be one in which eachcitizen developed a real split personality, acting selfishlyin the market place and altruistically at the ballot box.

[Meade, 1973]


The Fundamental Theorems of Welfare Economics

Theorem

First Fundamental Theorem of Welfare Economics

If markets exist for all goods and all agents are perfectly competitive price takers thenthe general equilibrium of a system of markets is a Pareto-efficient allocation ofresources.

Theorem

Second Fundamental Theorem of Welfare Economics

If the assumptions for the First Fundamental Theorem are satisfied, all preferences andtechnologies are convex and appropriate lump sum redistribution of endowments canbe achieved, then any Pareto-efficient allocation can be achieved as a generalequilibrium of a system of perfectly competitive markets.

Comment: The assumptions required are very strong, plus Pareto-efficiency isonly a bare minimum requirement for a socially desirable allocation. However, itis noteworthy that the FFTWE and SFTWE can hold for both selfish andaltruistic preferences (provided, as we shall see, altruistic preferences take anon-paternalistic form so that no externalities/missing markets are created).


The Edgeworth Box


The Edgeworth Box


The Edgeworth Box


The Edgeworth Box


The Edgeworth Box








The Non-Twisting Theorem

V1 =√

X1 +√

Y1 V2 =√

X2 +√

Y2

U1 = V1 + θ1V2 =√

X1 +√

Y1 + θ1

(√X2 +

√Y2

)U2 = V2 + θ2V1 =

√X2 +

√Y2 + θ2

(√X1 +

√Y1

)“Shrunken contract curve” lies between points d and c. Point a is a Pareto-efficientCGE. Point b represents a possible Pareto-inefficient CGE if redistributive bargaining isnot possible (e.g. due to a large number of each type of consumer).

→ X1 X2 ←

↑Y1

↓

Y2

a

b

c

d


The Rotten Kid Theorem and The Samaritan’s Dilemma

Theorem

Rotten Kid Theorem: All members of a family will behave efficiently, even if they arecompletely selfish (or imperfectly altruistic) provided that the head of the family issufficiently altruistic to make an operative transfer [Becker, 1974] [Bergstrom, 1989].

Consider the decision of a “rotten kid” over whether to take an action thatincreases or decreases their pre-transfer income at the expense or gain of anotherfamily member. The head of household will take into account the decision madeby the rotten kid when deciding how big a transfer of resources to give him.Any action which increases the overall collective family wealth therefore makesthe rotten kid better off. The rotten kid thus behaves fully efficiently.

The key assumption is that the actions taken by the rotten kids must be of sucha nature that they cause a shift rather than a change in the slope of the family’sutility possibilities frontier.

When “rotten kids” can take actions that distort the slope of the utilitypossibilities frontier, they may take inefficient actions from the perspective ofthe family. One of the main situations in which this condition does not apply iswhen there is a moral hazard problem between the head of the family and therotten kid due to certain goods in the rotten kid’s utility function not beingunder the direct control of the head of household via the transfer process. Thisis also known as the “Samaritan’s Dilemma” [Bruce & Waldman, 1990][Andreoni, 1989].


Intrinsic and Extrinsic Incentives

Intrinsic Incentives - Altruistic preferences provide anintrinsic motivation for individuals to exhibit altruisticbehaviour.

Extrinsic Incentives - Punishment systems provide anextrinsic motivation.

Often it is empirically difficult to distinguish between the two(e.g. enlightened self-interest in the repeated prisoners’dilemma) [Hammond, 1975].

These two forms of incentives represent alternative “socialtechnologies” that can potentially be used to achieve sociallybeneficial outcomes, but which can interfere with one anotherin a perverse manner. The moral preferences and institutionswhich have evolved in human society represent a particular“policy mix” which may (or may not) be socially optimal.


Punishment in Dynamic Games

[W]hen altruism improves static non-cooperativeoutcomes, it lessens the severity of credible punishments.An altruist may well be perceived as a “softy” and histhreats may not be taken seriously.

[Bernheim & Stark, 1988]

[T]he most efficient way to provide low payoffs, in termsof incentives to cheat, is to combine a grim present witha credibly rosy future.

[Abreu, 1986]


Punishment in Dynamic Games

[I]n comparison with a situation wherein altruism is absent altogether, theprevalence of just some altruism could result in Pareto inferior outcomes.Hence, if the formation of altruism may not only fail to do any good butmay actually make things worse whereas the formation of sufficiently highlevels of altruism is almost always beneficial,...a troubling discontinuityarises: to the extent that the formation of altruism is like the rising ofbread dough (i.e. it has to be gradual) groups yearning to build up theirsocial stock of altruism may have to endure Paretial deterioration beforeexperiencing Paretial gains. Perhaps one reason why a great manysocieties consist of self-interested economic men and women rather thanaltruistic economic men and women has to do with this nonmonotonicity.

[Stark, 1989]

Theorem

As altruism becomes perfect, the Nash equilibrium outcome in arepeated game becomes arbitrarily close to the socially efficientoutcome [Bernheim & Stark, 1988].


The Sequential Punishment Model - A Word on Idiom

The sequential punishment model presented in this paper isintended as a highly abstract and stylized representation of socialinteraction, rather than as a realistic model of a specific situation.A simple “parable” can often help with the intuition. Models witha similar idiom include:

Robinson Crusoe economy [Ruffin, 1972].

Samuelson’s “chocolate pension game” [Samuelson, 1958].

Diamond’s model of fiat money in a “coconut economy”[Diamond, 1984].

So, in that spirit, a desert island parable seems appropriate...


The Sequential Punishment Model - An Island Parable

Individuals (who have been on the island long enough to set up a “back garden”) finish off a cold beer one at atime and must decide whether to walk to the bin or just throw their bottle into another individual’s garden:

Width of island (= 1)

Cost of bottle landing (= 1)

Distance to bin(= πt ∈ [0, 1])

Discount factor - 0 ≤ δ < 1

Coefficient of altruism - θ < 1


The Sequential Punishment Model - Players’ Preferences

Felicity represents “private utility” from “economicallyfundamental” goods.

In period t, player t moves so as to maximize his expectedsocial utility ut , discounted looking forward:

Utility thus includes broader “moral preferences”. This is ofcourse only one among a number of alternative ways to definealtruism. The advantage is that it enables us to simplify awayfrom any “multiplier effects”. Not to say that these do notexist and are not important in the real world, but in thesequential punishment model we wish to focus on the role ofpunishment and its interaction with altruistic preferences in asclean and simple an environment as possible.

Social welfare function is utilitarian in felicities, or we canargue that Pareto efficiency requires an equilibrium where nobottles are thrown.


The Sequential Punishment Model - Three Effects

Temptation Effect - Individuals with higher altruism are lesstempted to inflict harm upon another individual for their owngain. (This is the main benefit from higher altruism.)

Willingness Effect - Individuals with higher altruism are lesswilling to punish another individual for a previousmisdemeanour by inflicting harm upon them. (This is a costto higher altruism.)

Severity Effect - Individuals with higher altruism also findsome kinds of punishment less severe. In particular, if a finewas imposed, and some or all of the revenue is redistributedto another individual whose felicity has some weight in theutility function of the individual we are trying to punish, thenany given size of fine is less severe for the punishee. (Anothercost to higher altruism.)


The Sequential Punishment Model - Notation

δ - Discount factor.

θ - Coefficient of altruism.

πt ∈ [0, 1] - Benefit from harming / punishing in period t(randomly distributed between π and 1).

θ∗ - Socially optimal level of altruism - Enables efficientequilibrium to be sustained for largest possible range of δ.

δ∗ - Lowest possible value of δ for which the socially efficientoutcome can be sustained. (Corresponds to θ∗.)

κ(θ) - Net loss in utility when deviating from socially efficientequilibrium when optimal punishment is applied.


Overview - Socially Efficient Equilibria

A - Supportable usingNash-reversion path

B - Supportable usingoptimal path


The Sequential Punishment Model - Solution Concepts

Folk Theorem - [Aumann & Shapley, 1992][Rubinstein, 1979] [Fudenberg & Maskin, 1986] For any givenθ, as δ −→ 1, the socially efficient outcome becomessupportable. We are interested here, however, in whathappens as θ −→ 1 for any given δ < 1.

Optimal Penal Codes - [Abreu, 1988] Abreu’s framework ofoptimal penal codes in the form of punishment paths providesa natural framework that can be adapted to analyse sociallyefficient equilibria in the sequential punishment model.

Renegotiation Proofness - [Farrell & Maskin, 1989][Benoit & Krishna, 1993] We assume that society is able toavoid the temptation to let malefactors “off the hook”. Thuswe stick with subgame perfection rather than further refiningthe equilibrium criterion.


Results - Illustrated Using Uniform Distribution of Benefit

abcdef

g

-0.5

0

0.5

1

1.5

0 0.2 0.4 0.6 0.8 1

θ

κ(θ)

Theorem

As θ −→ 1−, κ (θ) −→ 0−.

Proof.

Intuition: As θ −→ 1−, the willingness and severity effect become negligible, and thetemptation effect ensures that dκ

dθ> 0.


Results - Illustrated Using Uniform Distribution of Benefit

abcdef

g

-0.5

0

0.5

1

1.5

0 0.2 0.4 0.6 0.8 1

θ

κ(θ)

Theorem

θ∗ ∈ (0, 1) ( θ∗ = 1− 1√3≈ 42% for uniform benefit distribution)

Proof.Intuition: Let δ = δ∗ = θ∗. If θ = θ∗ + ε then willingness and severity effect dominatetemptation effect, so dκ

dθ< 0. If θ < θ∗ then (because punishment is maximal)

willingness effect is 0, temptation effect dominates severity effect, so dκdθ

> 0.


Other Benefit Distributions

Figure: Socially efficient equilibria for g(π) = 1, g(π) = 2π and g(π) = 3π2.

Figure: Socially efficient equilibria for g(π) = 4π3, g(π) = 5π4 and g(π) = 6π5.


Multilevel Selection Theory


Haystacks Model

Note: Although the Haystacks model provides a neat mathematicalframework for analysing group selection, there are othermechanisms in play that have a similar effect and make groupselection a lot more plausible for cultural evolution:

Ostracism

Inter-group conflict


Haystacks Model

Source: [Cooper & Wallace, 2004].A constant positive probability of mutations rules out Type Icycles, and we then need the number of periods of isolation to bein a “Goldilocks zone” to ensure Type II cycles and enable altruismto survive in the population.


Evolutionary Sequential Punishment Model

A’s Phenotype H L

B’s Phenotype θA ≥ π θA < π

H θB ≥ π

0

0

0

π

π − 1

−12

L θB < π

−12

π − 1

π

π2

π − 1

π2

Middle payoff is social welfare (sum of payoffs of players 1 to 3). Player A and B facea 50-50 lottery on whether they get to move first as player 1 or second as player 2.With universal high altruism, neither player ever inflicts harm. With universal lowaltruism, player 2 inflicts harm on player 3 but the credible threat of “switching” thisinto player 1 deters player 1 from inflicting harm. If player 1 is altruistic and player 2 isselfish then player 1 does not inflict harm but player 2 does (upon player 3). If player1 is selfish and player 2 is altruistic then player 2 does not inflict harm so player 1 does(upon player 2).


Evolutionary Sequential Punishment Model

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 50 100 150 200 250 300 350 400 450 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 50 100 150 200 250 300 350 400 450 500

t

tU

q

Grey line shows altruism in sequential punishment model, black line in analogous prisoners’ dilemma model.

Figure: 52 periods of isolation, π = 0.075

Theorem

Altruism always survives more easily in the prisoners’ dilemma thanin the sequential punishment model [Povey, 2014].


Policy Consequences

The impact of the presence of altruism has been analysed in the context of thetheory of cost-benefit analysis. It has been shown that in the presence ofnon-paternalistic altruism, household willingness to pay for a public goodexceeds the sum of individuals’ willingness to pay [Quiggin, 1998]. An attemptto estimate the results of the presence of paternalistic altruism on the value ofstatistical life by calibrating to 2.5 individuals per altruistic family predicted thatthis value is 10%-40% higher than the individual value [Jones-Lee, 1992].

A number of overlapping generations models of environmental degradation haveshown that the presence of partial altruism does not guarantee an efficientinternalization of these externalities [Jouvet et al., 2000] [Turner, 1997]. It hasalso been found that co-operation between nations to internalise currentenvironmental externalities may lead to a deterioration of future environmentrelative to non-co-operation because improved environmental technology freesup more resources for consumption [John & Pecchenino, 1997].

It has been shown that altruism can either increase or decrease optimal Pigoviantaxation depending on its precise form [Johansson, 1997]. The optimal subsidyfor voluntary giving has also been found to depend in a dramatic way upon thenature of the altruistic motivation for giving [Kaplow, 1998].

It has been shown that infectious happiness (similar to altruism if agents areconscious of its effects) reduces market failures such as over-supply of labour(due to relative income effects) and under-supply of public goods [Povey, 2015].


References I

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Abreu, Dilip (1988).

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References III

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References V

Johansson, Olof (1997).

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References VI

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“Punishment and the potency of group selection”.Journal of Evolutionary Economics, 24(4), 799–816.

Povey, Richard (2015).

“The welfare economics of infectious happiness”.Economics Letters, 133, 1 – 3.

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The Limits to Altruism - University of Oxfordusers.ox.ac.uk/~sedm1375/GPIpresentation.pdf · The Limits to Altruism Richard Povey Hertford College and St Hilda’s College, University